The mass spectrometer is a known instrument for measuring the gas-phase mass ions or molecular ions in a vacuum chamber via ionizing the gas molecules and measuring the mass-to-charge ratio of the ions. One specific type of mass spectrometer is the ion trap mass spectrometer. The quadrupole ion trap was first described in U.S. Pat. No. 2,939,952 by Paul and H. Steinwedel, where the disclosed ion trap is composed of a ring electrode and a pair of opposite end cap electrodes. The inner surfaces of the ring and two end cap electrodes are rotationally symmetric hyperboloids.
The quadrupole ion trap and another type of mass spectrometer—the quadrupole mass filter both utilize the stability or instability of ion trajectories in a dynamical electric field to separate ions according to ions' mass-to-charge ratios—m/Q. As is known in the art, the ion movement inside the quadrupole field can be derived from Mathieu equation. Stability diagram is utilized to determine an ion's stable or instable movement in the quadrupole field. Theories and applications of quadupole mass filter and quadrupole ion trap are described in numerous literatures such as “Quadrupole Mass Spectrometry”, edited by P. H. Dawson, Elsevier, Amsterdam, 1976; “Quadrupole Storage Mass Spectrometry”, by R. E. March and R. J. Hughes, John Wiley & Sons, New York, 1989; “Practical Aspects of Ion Trap Mass Spectrometry”, Volumes I, II and III edited by R. E. March and John F. J. Todd, CRC Press, Boca Raton, New York, London, Tokyo, 1995, to name a few.
In cylindrical coordinates (r, z) (since the field is rotationally symmetric), an ideal or pure three-dimensional quadrupole potential distribution Φq is expressed asΦq=Φ0/R02*(r2−2*z2)  (1)where R0 is a parameter of length dimension. Φ0 is a position-independent factor which is time dependent. The hyperboloid metallic electrode surfaces of Paul trap is shaped by equipotential surfaces of equation (1) withΦq=+1 and −1; Φ0=1; and R0=r0;where r0 is the distance from the center of the trap to apex of the ring electrode. The distance between apexes of two opposite caps is 2*z0. When an RF (radio frequency ) voltage having magnitude V and frequency Ω, and a DC (direct current) voltage having magnitude U are applied to the ring electrode where two caps are grounded, ions can be trapped in the generated RF electric quadrupole field. It is well-known that the movement of an ion having mass m and electric charge Q inside an ideal RF quadrupole field can be derived from the following Mathieu equation:d2u/dξ2+(au−2*qu*cos(2*ξ))*u=0  (2)Where u=r, z; ξ=Ω*t/2; au=−8*e*U/(m*r02*Ω2); qu=4*e*V/(m*r02*Ω2).
The Mathieu equation (2) can be solved using analytical methods. The fundamental properties of the ion movement are as follows:                1. The ion movement in the axial or z direction is completely decoupled from the movement in the direction perpendicular to the z-axis, normally called the r direction.        2. The RF field intensity is linear in the r and z directions of the cylindrical coordinates and has only one parameter describing the periodicity.        3. The stability of ions of a given mass-to-charge ratio in an infinitely large quadrupole field does not depend on the initial movement conditions of the ions, it depends on the field parameters.        4. Only the two “mass-related amplitude parameters” au for DC field and qu for RF field determine whether the oscillation amplitude of the ions will increase to infinity without limit. This is described by the well known “stability diagram” for quadrupole ion traps.        5. If the set of parameters (au, qu) is kept inside the stability region of the stability diagram, the ions will perform stable oscillation in the r and z directions at certain frequencies—the so-called secular frequencies. The fundamental secular frequency is ½*βu* Ω and the parameter βu is a value dependent on the parameters au, qu. The iso-βr, and iso-βz lines subdivide the stability region.        6. The frequencies of the secular oscillation are independent of the ion oscillation's amplitude.        
A mass spectrum can be obtained by the so-called mass scanning method in an ion trap mass spectrometer. Dawson and Whetten in U.S. Pat. No. 3,527,939 described a “mass-selective storage” method. The method is based on the same quadrupole mass filter operating principle, namely only ions with a particular mass-to-charge ratio m/Q possess stable movement trajectories and are selectively stored in the trap along with a set of parameters (au, qu) which lie in the apex of the first stability region of the stability diagram. The ions are extracted to detector by a pulse on an end cap electrode after certain time period. A mass spectrum is obtained by swapping or scanning slowly DC and RF voltages at constant U/V. Ions of different mass-to-charge ratios are ejected through one or a plurality of holes on the center of an end cap and are detected by an ion detector, such as a secondary electron multiplier, sequentially or one mass-to-charge-ratio ion after the other.
Stafford, Kelley and Stephens described another mass scanning method “mass-selective instability” in U.S. Pat. No. 4,548,884, where only RF voltage is applied to ring electrode and ions with a range of different mass-to-charge ratios are trapped. The RF voltage is swept increasingly with time. When the related parameter qz approaches the boundary of the first stability region (e.g., az=0, qz=0.908), oscillations of the ions of a particular m/Q, with that parameter, will be unstable in z direction and be ejected. A mass spectrum is obtained by scanning RF voltage and detecting the unstable ions of different m/Q sequentially.
Another mass scanning method of obtaining a mass spectrum is the mass-selective resonance ejection method described by Syka, Louris, Kelley, Stafford and Reynolds in U.S. Pat. No. Re 34,000. The method employs an auxiliary AC (alternating current) voltage which is applied between the caps. When the RF voltage is swept increasingly with time, the oscillating secular frequency of trapped ions of a particular m/Q will increase correspondingly. When the frequency of the AC voltage coincides with the secular frequency of the ions, the ions will be oscillated in resonance and be ejected eventually. The resonance is linear because the amplitude of the oscillation is independent of the frequency according to Mathieu equation (1). The method also is utilized in a linear two-dimensional quadrupole ion trap described by Bier et al, in U.S. Pat. No. 5,420,425.
All above mentioned ion traps used the conventional Paul's trap structure with two caps and one ring. They are generally operated in a high or medium high vacuum condition. However, if the ion traps are operated in a lower vacuum, the linear resonance frequency curve will be broadened due to massive collision between ion and neutral gas, which will cause the mass resolving power to decrease dramatically.
Another issue is that, even with precisely shaped trap-electrodes, the field inside the practical Paul ion traps demonstrates unavoidable deviates from the ideal quadrupole field due to a wide variety of factors such as the truncation to finite size, holes on the caps if no special corrections are applied etc. Deviation of electrode shapes from pure quadrupole systems result in the superposition of higher multipole fields, like hexapole, octopole onto the quadrupole field. These non-linear components of the field may be introduced either from electrode faults or by deliberate superposition.
The general potential distribution Φ having rotational symmetry within a boundary is expressed in spherical coordinates (ρ,θ) as follows:Φ(ρ,θ)=Φ0*Σ(An*ρn/r0n*Pn(cos θ))  (3)where n is integers from zero to infinity, Σ is the sum, An are weight factors which are determined from the boundary condition of the trap, Pn(cos θ) are Legendre polynomials of order n. In ion trap mass spectrometer, Φ is a position-independent but time-dependent quantity representing the strength of the potential, Φ0=Φ0(t). Because Φ0 is time-dependent, the potential including higher multipoles is a dynamic or time-dependant potential and corresponding field is a time-dependant field. A ideal three-dimensional quadrupole field Φq is described by n=2 and A2=−2 (An=0 if n is not equal to 2) in Eq. (3):Φq=−2*Φ0/r02*ρ2*P2(cos θ))=Φ0/r02*(r2−2*z2)  (4)which is the same as Eq. (1). The different terms of the sum in Eq. (3) constitute the “multipole components” of the potential distribution. A few of exemplary lowest multipoles are:    n=3 (hexapole):Φh=A3*Φ0/r03*ρ3*P3(cos θ))=A3*Φ0/r03*(2*z3−3*z*r2)/2  (5)    n=4 (octopole):Φo=A4*Φ0/r04*ρ4*P4(cos θ))=A4*Φ0/r04*(8*z4−24*z2*r2+3*r4)/8  (6)    n=5 (decapole):Φde=A5*Φ0/r05*ρ5*P5(cos θ))=A5*Φ0/r05*(8*z5−40*z3*r2+15*z*r4)/8  (7)    n=6 (dodecapole):Φdo=A6*Φ0/r06*ρ6*P6(cos θ))=A6*Φ0/r06*(16*z6−120 *z4*r2+90*z2*r4−5*r6)/16  (8)    If n=1 (An=0 for n≠1, it is dipole potential:Φd=A1*Φ0/r0*ρ*P1(cos θ)=A1*Φ0/r0*z  (9)
where A1, A2, A3, A4, A5 and A6 are weight factors of the corresponding filed components, which are determined from the boundary condition or the tape structure. For example, for hyperboloid boundary with infinitively-length, which corresponds to that the weight factor A2 equals to −2 (An=0 if n is not equal to 2), the ideal or pure quadrupole will be obtained.